Let be a tree rooted at endowed with a nearest-neighbor transition probability that yields a recurrent random walk. We show that there exists a function biharmonic off whose Laplacian has potential theoretic importance and, in addition, has the following property: Any functiononwhich is biharmonic outside a finite set has a representation, unique up to addition of a harmonic function. We obtain a characterization of the functions biharmonic outside a finite set whose Laplacian has 0 flux similar to one that holds for a function biharmonic outside a compact set in Rfor= 23, and 4 proved by Bajunaid and Anandam. Moreover, we extend the definition of flux and, under certain restrictions on the tree, we characterize the functions Biharmonic outside a finite set that have finite flux in this extended sense.